常用数学求和公式

常用数列求和

\[\sum_{i=1}^ni = 1 + 2 + 3 + ... + n = \frac{n(n+1)}{2} \]

\[\sum_{i=1}^ni^2 = 1^2 + 2^2 + 3^2 + ... + n^2 = \frac{n(n+1)(2n+1)}{6} \]

\[\sum_{i=1}^ni^3 = 1^3 + 2^3 + 3^3 + ... + n^3 = \frac{n^2(n+1)^2}{4} \]

\[\sum_{i=1}^ni(i+1) = 1\times 2 + 2\times 3 + 3\times 4 + ... + n \times (n+1) = \frac{n(n+1)(n+2)}{3} \]

\[\sum_{i=1}^n\frac1{i(i+1)} = \frac1{1\times 2} + \frac1{2\times 3} + \frac1{3\times 4} + ... + \frac1{n \times (n+1)} = \frac{n}{n+1} \]

等差数列求和

\[S_n = \frac{(a_1+a_n)n}{2} = \frac{d^2}{2}n^2+\big(a_1-\frac d2\big)n \]

等比数列求和

\[S_n = a_1\times \frac{1-q^n}{1-q}=\frac{a_1-a_n\times q}{1-q} (q \ne 1) \]

等差乘等差求和

等差×等差展开后可得一个关于 \(n\) 的二次三项式 \(An^2+Bn+C\)，对每一项拆开套用求和公式即可，即：

\[\sum_{i=1}^nAn^2+Bn+C=\frac{An(n+1)(2n+1)}{6}+\frac{Bn(n+1)}{2}+Cn \]

等差乘等比求和

对于 \(h_i=(an+b)\cdot q^{n-1}\), 有:

\[\sum_{i=1}^nh_i=(An+B)q^n-B,其中A=\frac a{q-1},B=\frac{b-A}{q-1} \]